Pole-free solutions of the first Painlev\'e hierarchy and non-generic critical behavior for the KdV equation

TL;DR

This paper proves the existence of pole-free solutions for all even members of the Painlevé I hierarchy, analyzes their asymptotics, and links them to non-generic critical behavior in the small dispersion limit of the KdV equation.

Contribution

It extends the understanding of pole-free solutions to the Painlevé I hierarchy and their role in describing non-generic critical phenomena in the KdV equation.

Findings

Existence of real pole-free solutions for all even Painlevé I hierarchy members

Asymptotic descriptions of these solutions

Application to non-generic critical behavior in KdV equation

Abstract

We establish the existence of real pole-free solutions to all even members of the Painlev\'e I hierarchy. We also obtain asymptotics for those solutions and describe their relevance in the description of critical asymptotic behavior of solutions to the KdV equation in the small dispersion limit. This was understood in the case of a generic critical point, and we generalize it here to the case of non-generic critical points.